Hyperreal Intermediate Value Theorem

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SUMMARY

The discussion centers on the application of the Intermediate Value Theorem (IVT) within the context of hyperreal numbers. The user references "Elementary Calculus: An Infinitesimal Approach" by Jerome Keisler, specifically chapter 3, which provides a proof of the theorem's applicability to hyperreals. The inquiry focuses on whether a continuous function defined on hyperreals maintains the IVT for closed intervals [a,b], where a and b may be unlimited. The user expresses uncertainty about the topology of hyperreals and the limitations of Los's theorem in this context.

PREREQUISITES
  • Understanding of the Intermediate Value Theorem
  • Familiarity with hyperreal numbers and their properties
  • Knowledge of topology, particularly in relation to continuous functions
  • Basic concepts of infinitesimals as presented in "Elementary Calculus: An Infinitesimal Approach"
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  • Study the proof of the Intermediate Value Theorem in "Elementary Calculus: An Infinitesimal Approach" by Jerome Keisler
  • Research the topology of hyperreal numbers and its implications for continuous functions
  • Explore the limitations and applications of Los's theorem in non-standard analysis
  • Investigate further into continuous functions defined on hyperreal intervals
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Mathematicians, students of calculus, and anyone interested in non-standard analysis and the properties of hyperreal numbers.

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I haven't thought too much about this, but it seems to me that the intermediate value theorem would transfer. Am I incorrect?
 
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conscipost said:
I haven't thought too much about this, but it seems to me that the intermediate value theorem would transfer. Am I incorrect?

In Elementary Caculus: An Infinitesimal Approach by Jerome Keisler in chapter 3, there is good proof of this.
The chapters and whole book is a free down-load HERE.
 
Plato said:
In Elementary Caculus: An Infinitesimal Approach by Jerome Keisler in chapter 3, there is good proof of this.
The chapters and whole book is a free down-load HERE.

This was not quite what I was thinking about. But I think it helps clarify the question I had. I have not thought much about the topology of the hyperreals but what I'm asking is:

Say I have a continuous function defined on the hyperreals and I take a "closed" interval [a,b], that is the set of all hyperreals between a and b, where a and b may very well be unlimited, does it follow that for any c: f(a)<c<f(b) there exists an x so that f(x)=c.

The star transform of [a,b] wouldn't give unlimited elements as far I'm looking into it, so I'm afraid this would have to be proven using something other than Los's theorem.
 

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